115 
1919-20.] On a Class of Graduation Formulae. 
Since the sums of the odd powers over all the values from — m to +m 
vanish, it is obvious that only half of the “ normal equations ” will contain 
c 0 . Denoting ( — m) r + ( — m+l) r + . . . +( — l) r + (l) r -f . . . + (m — l) r + m r 
by for all values of r except zero, the normal equations containing c 0 are 
Vo "1“ ^2 C 2 "t ^4^4 + . . . + — V 
^ 2^*0 ^ 4 C 2 ^ 6^4 + • • • + ^ 2 &+ 2 < - : 2 fc “ 
V c o + V+ 2 C 2 + V+ 4 C 4 + • • • + V c 2& = X 2k u s , where j = 2k or 2k + 1 , 
2> 0 = 2m + 1, and %s p u s = ( - m) p u~ m + ( - m + l)^_ m+1 + . . . + (m - 1 )% m _ 1 + m v u m . 
Solving these equations determinan tally, we have as the value of c 0 , 
which is equal to u\ 0), the graduated value of : 
~%U S %S 2 U s %S*U s 
S 2 s 4 s 3 
s* s 6 s 8 
V+2 
^2&+4 
^0 
^2 
^4 
^8 
*2fc 
V+2 
^2^+4 
(A), 
^2&+2 
X 
2&+4 
a linear combination of the quantities Xu s , Hs 2 u s . . . 2-s 2k u s . 
If it be supposed that each of the given data is weighted, we may take 
W-m , w-rn+ 1 , w -m +2 • • • w -i • • • w m-\ ■> to he the weights, respectively, 
of w_ w , u_ m+1 , u _ m+ 2 . . . , u 0 , . . . w m _j , ; the normal equation in c 0 
will be 
c 0 {w- m +w-m+i+ • . • +w m } +c 1 {-rnw-m~{rn-l)w- m+ i- . . . + (m-l)w m -i+mw m } 
.... +c 2 {m 2 itf_ m + (m- lfw- m+1 +. . . + rn 2 w m }-j- . . . 
= W- m U- m + W- m+ iU- m+ i+ . . . +w m u m . 
It follows that when w m = -Wm, etc., the quotient of the two determinants (A) 
will still represent the graduated value of u 0 , provided the notation in the case of 
weighted data is understood to be 
3, 0 = W- m + W- m +i + . . . + W 0 + . . . +W m -i + W m . 
2r = {-m) r w-. m + (-m + l) r w- m +i+ . . . +{m-l) r w m -i + ?n r w . 
'2l( s = W-mW-m + W-m+l'U-m+lk- . • • +W 0 U Q -r . . . + + W m U m . 
'ZsPus = ( - + ( - m + 1 ) p w- m+ iU - m + i + • . . +[m- 1 ) p u m -\w m -\ + mSu m w m . 
